Just as the difference between two sample means is normallydistributed for large
ID: 2913775 • Letter: J
Question
Just as the difference between two sample means is normallydistributed for large samples, so is the difference between twosample proportions. That is if Y1 and Y2 are independent binomialrandom variables with parameters (n1 , p1) and (n2, p2),respectively, then (Y1 / n1) - (Y2 / n2 ) is approximatelynormally distributed for large values of n1 and n2. Find V( Y1 / n1 - Y2 /n2 ) Just as the difference between two sample means is normallydistributed for large samples, so is the difference between twosample proportions. That is if Y1 and Y2 are independent binomialrandom variables with parameters (n1 , p1) and (n2, p2),respectively, then (Y1 / n1) - (Y2 / n2 ) is approximatelynormally distributed for large values of n1 and n2. Find V( Y1 / n1 - Y2 /n2 )Explanation / Answer
given that Y1~ B(n1,p1)=> E(Y1) = n1p1 andV(Y1) = n1p1q1 and Y2 ~ B(n2,p2)=> E(Y2) = n2p2 andV(Y2) = n2p2q2 V(Y1/n1 - Y2/n2) =V(Y1/n1 )+ V(Y2/n2 ) -2Cov( Y1/n1 ,Y2/n2) = (1/n21 )V(Y1)+ (1/n2 2 ) V(Y2) - 0(since the Y1 and Y2 are independent) = (1/n21 )n1p1q1+(1/n2 2 )n2p2q2 V(Y1/n1 -Y2/n2) = (p1q1/n1 ) + (p2q2/n2 ) V(Y1/n1 - Y2/n2) =V(Y1/n1 )+ V(Y2/n2 ) -2Cov( Y1/n1 ,Y2/n2) = (1/n21 )V(Y1)+ (1/n2 2 ) V(Y2) - 0(since the Y1 and Y2 are independent) = (1/n21 )n1p1q1+(1/n2 2 )n2p2q2 V(Y1/n1 -Y2/n2) = (p1q1/n1 ) + (p2q2/n2 )Related Questions
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